Search

Given a compact and connected metric space (continuum) X, we study topological and dynamical properties of the hyperspace of normal fuzzy sets F1(X) equipped with the Hausdorff, endograph or sendograph metric. Among the many results we show that it is contractible, path connected, locally contractible, locally path connected, locally simply connected and locally connected. For the endograph metric the hyperspace F1(X) is a continuum, and then for a topological graph X we show how, using the inverse limit approach of Barge and Martin, the inverse limit of a fuzzy dynamical system on X can be realized as an attractor of a fuzzy dynamical system on a manifold.

The goal of this paper is to establish connections between two entropy concepts, namely the fuzzy entropy of a fuzzy set, which appears in fuzzy set theory, and the topological entropy of Zadeh's extensions of continuous maps defined on metric compact spaces, which is a topological dynamics concept. We develop several concepts that allow us to simultaneously address the two entropy concepts. Except for stating several basic properties, we are partially capable of specifying in which classes of fuzzy sets dynamically complex behavior can appear by using these instruments. Such instruments are necessary for a proper understanding of the behavior of fuzzy systems generated from crisp, discrete dynamical systems.

In this paper we provide several results concerning the approximation of Zadeh's extension of a given function. We provide several simple observations due to which we provide a general procedure allowing to approximate Zadeh's extension of any continuous map. Among other things, we demonstrated that the technique of F-transform can also be used to approximate this extension. Further, we studied how the choice of the metric on the space of fuzzy sets (resp. fuzzy numbers) can affect the quality of the approximation, and we clarified that, in comparison to recent habits, the choice of the levelwise metric need not be the most appropriate one, and that the approximation process could be simplified. Our results are general in the sense that we mostly require nothing but the continuity of the original map.

In the main result of this article, we prove that the topological entropies of a given interval map and its Zadeh's extension (fuzzification) to the space of fuzzy numbers (i.e., the space of fuzzy sets with connected α-cuts) are the same. This result is in contrast with our previous result, which claimed that the topological entropy of the Zadeh's extension significantly increases for a majority of simple interval maps. In addition, we prove some properties of the limit sets of trajectories that are generated by iterating the fuzzy set valued function on connected fuzzy sets; for instance, we specify the shapes of the possible limit sets. Furthermore, the presented topics are studied for set-valued (induced) discrete dynamical systems. The main results are proved with a variational principle that describes the relations between topological entropy and measure-theoretical entropy.

It is well known that any given discrete dynamical system uniquely induces its fuzzified counterpart, i.e. a discrete dynamical system on the space of fuzzy sets. In this paper we study relations between dynamical properties of the original and fuzzified dynamical system. Especially, we study conditions used in the definition of Devaney chaotic maps, i.e. periodic density and transitivity. Among other things we show that dynamical behavior of the set-valued and fuzzy extensions of the original system mutually inherits some global characteristics and that the space of fuzzy sets admits a transitive fuzzification. This paper contains the solution of the problem that was partially solved by Roman-Flores and Chalco-Cano [Some chaotic properties of Zadeh's extension, Chaos, Solitons and Fractals 35(3) (2008) 452-459].

A discrete dynamical system is given by a compact metric space X and any continuous self-map defined on X. This discrete dynamical system can be naturally extended to the space of fuzzy sets on X. In this paper we study relations between the sizes of the topological entropies of the original dynamical system and of its fuzzy counterpart. Among other things, we present a constructive proof of the fact that even very weak assumptions on the crisp discrete dynamical system ensure infinite topological entropy of the fuzzy system. However, we also show that there are subsystems of the fuzzy dynamical system with topological entropy equal to that of the crisp dynamical system.